I'm trying to solve the following problem, but I'm not sure if it's actually possible/true:
"Let $M$ be a closed pure lambda term without normal form. Prove that $\lambda x.\lambda y.(M (x y))$ has no normal form either".
The similar problem with $\lambda x.(M x)$ can be proved trivially by eta-reduction; does this automatically imply the proof of my problem?
If $\lambda x.\lambda y.(M (x y))$ can have a normal form, can you give a counterexample expression $M$?